WELCOME [] Theory.pdfOutline 1 Set Theory Introduction to Sets Sets 2 Origin of Set Theory 3 De...

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WELCOME

J.Maria Joseph Ph.D., SJC, Trichy-2. Set Theory July 1, 2015 1 / 62

Set Theory

J. Maria Joseph Ph.D.,

Assistant Professor, Department of Mathematics,St. Joseph’s College, Trichy - 2.

July 1, 2015

Outline

1 Set TheoryIntroduction to SetsSets

2 Origin of Set Theory

3 Definitions

4 Problems

Set Theory

This is where mathematics starts.

Introduction to Sets

What is set ?Well, simply put, it’s a collection.

DefinitionA set is a collection of well defined objects or things.

First we specify a common property among “things“ andthen we gather up all the “things“ that have thiscommon property.

First we specify a common property among “things“

andthen we gather up all the “things“ that have thiscommon property.

For ExampleThe items you wear: shoes, socks, hat,shirt, pants, and so on.

I’m sure youcould come up with at least a hundred.This is known as a set.

For ExampleThe items you wear: shoes, socks, hat,shirt, pants, and so on. I’m sure youcould come up with at least a hundred.

This is known as a set.

For ExampleThe items you wear: shoes, socks, hat,shirt, pants, and so on. I’m sure youcould come up with at least a hundred.This is known as a set.

For ExampleTypes of fingers.

This set includesindex, middle, ring, and pinky.

So it is just things grouped together with a certainproperty in common.

For ExampleTypes of fingers. This set includesindex, middle, ring, and pinky.

NotationThere is a fairly simple notation for sets.

We simply listeach element, separated by a comma, and then put somecurly brackets around the whole thing.

{3, 6, 91, . . .}The curly brackets { } are sometimes called “setbrackets“ or “braces“.

NotationThere is a fairly simple notation for sets. We simply listeach element, separated by a comma, and then put somecurly brackets around the whole thing.

{3, 6, 91, . . .}

The curly brackets { } are sometimes called “setbrackets“ or “braces“.

Notation for Examples{ socks, shoes, watches, shirts, . . . } - For Example 1

{ index, middle, ring, pinky } - For Example 2

Notice how the first example has the “ · · · “. The threedots · · · are called an ellipsis, and mean “continue on“.

The first set { socks, shoes, watches, shirts, . . .} we callan infinite set,the second set { index, middle, ring, pinky } we call afinite set.

Notation for Examples{ socks, shoes, watches, shirts, . . . } - For Example 1{ index, middle, ring, pinky } - For Example 2

Notice how the first example has the “ · · · “.

The threedots · · · are called an ellipsis, and mean “continue on“.

The first set { socks, shoes, watches, shirts, . . .} we callan infinite set,

the second set { index, middle, ring, pinky } we call afinite set.

Numerical SetsSo what does this have to do with mathematics ?

Whenwe define a set, all we have to specify is a commoncharacteristic. Who says we can’t do so with numbers ?

For Example

� Set of even numbers: {. . . ,−4,−2, 0, 2, 4, . . .}� Set of odd numbers: {. . . ,−3,−1, 1, 3, . . .}� Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, . . .}� Positive multiples of 3 that are less than 10: {3, 6, 9}

Numerical SetsSo what does this have to do with mathematics ? Whenwe define a set,

all we have to specify is a commoncharacteristic. Who says we can’t do so with numbers ?

For Example

Numerical SetsSo what does this have to do with mathematics ? Whenwe define a set, all we have to specify is a commoncharacteristic.

Who says we can’t do so with numbers ?

For Example

Numerical SetsSo what does this have to do with mathematics ? Whenwe define a set, all we have to specify is a commoncharacteristic. Who says we can’t do so with numbers ?

For Example

� Set of even numbers: {. . . ,−4,−2, 0, 2, 4, . . .}

� Set of odd numbers: {. . . ,−3,−1, 1, 3, . . .}� Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, . . .}� Positive multiples of 3 that are less than 10: {3, 6, 9}

For Example

� Set of even numbers: {. . . ,−4,−2, 0, 2, 4, . . .}� Set of odd numbers: {. . . ,−3,−1, 1, 3, . . .}

� Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, . . .}� Positive multiples of 3 that are less than 10: {3, 6, 9}

For Example

� Set of even numbers: {. . . ,−4,−2, 0, 2, 4, . . .}� Set of odd numbers: {. . . ,−3,−1, 1, 3, . . .}� Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, . . .}

� Positive multiples of 3 that are less than 10: {3, 6, 9}

For Example

Universal SetAt the start we used the word “things“ in quotes.

Wecall this the universal set. It’s a set that containseverything. Well, not exactly everything. Everything thatis relevant to our question.

Universal SetAt the start we used the word “things“ in quotes. Wecall this the universal set.

It’s a set that containseverything. Well, not exactly everything. Everything thatis relevant to our question.

Universal SetAt the start we used the word “things“ in quotes. Wecall this the universal set. It’s a set that containseverything.

Well, not exactly everything. Everything thatis relevant to our question.

Universal SetAt the start we used the word “things“ in quotes. Wecall this the universal set. It’s a set that containseverything. Well, not exactly everything.

Everything thatis relevant to our question.

Universal SetAt the start we used the word “things“ in quotes. Wecall this the universal set. It’s a set that containseverything. Well, not exactly everything. Everything thatis relevant to our question.

Some More NotationWhen talking about sets, it is fairly standard to use

Capital Letters A,B ,C , . . . to represent the set, andlower-case letters a, b, c , . . . to represent an element inthat set.

For ExampleA = {a, e, i , o, u}

Here A denotes the set of vowels, and a, e, i , o, u is anelement of the set A.

Some More NotationWhen talking about sets, it is fairly standard to useCapital Letters A,B ,C , . . . to represent the set,

andlower-case letters a, b, c , . . . to represent an element inthat set.

Some More NotationWhen talking about sets, it is fairly standard to useCapital Letters A,B ,C , . . . to represent the set, andlower-case letters a, b, c , . . . to represent an element inthat set.

Some More NotationWhen we say an element a is in a set A,

we use thesymbol ∈ to show it. And if something is not in a setuse /∈.

For ExampleSet A is {1, 2, 3}. We can see that 1 ∈ A, but 5 /∈ A.

Some More NotationWhen we say an element a is in a set A, we use thesymbol ∈ to show it.

And if something is not in a setuse /∈.

Some More NotationWhen we say an element a is in a set A, we use thesymbol ∈ to show it. And if something is not in a setuse /∈.

For ExampleSet A is {1, 2, 3}.

We can see that 1 ∈ A, but 5 /∈ A.

For ExampleSet A is {1, 2, 3}. We can see that 1 ∈ A,

but 5 /∈ A.

EqualityTwo sets are equal if they have precisely the samemembers.

Now, at first glance they may not seen equal,so we may have to examine them closely!

EqualityTwo sets are equal if they have precisely the samemembers. Now, at first glance they may not seen equal,

so we may have to examine them closely!

EqualityTwo sets are equal if they have precisely the samemembers. Now, at first glance they may not seen equal,so we may have to examine them closely!

For ExampleAre A and B equal where:

� A is the set whose members are the first four positivewhole numbers

� B = {4, 2, 1, 3}

Let’s check. They both contain 1. They both contain 2.And 3, And 4. And we have checked every element ofboth sets, so: Yes, they are equal !

� B = {4, 2, 1, 3}

Let’s check.

They both contain 1. They both contain 2.And 3, And 4. And we have checked every element ofboth sets, so: Yes, they are equal !

� B = {4, 2, 1, 3}

Let’s check. They both contain 1.

They both contain 2.And 3, And 4. And we have checked every element ofboth sets, so: Yes, they are equal !

� B = {4, 2, 1, 3}

Let’s check. They both contain 1. They both contain 2.

And 3, And 4. And we have checked every element ofboth sets, so: Yes, they are equal !

� B = {4, 2, 1, 3}

Let’s check. They both contain 1. They both contain 2.And 3, And 4.

And we have checked every element ofboth sets, so: Yes, they are equal !

� B = {4, 2, 1, 3}

Let’s check. They both contain 1. They both contain 2.And 3, And 4. And we have checked every element ofboth sets,

so: Yes, they are equal !

� B = {4, 2, 1, 3}

SubsetsWhen we define a set, if we take pieces of that set, wecan form what is called a subset.

For ExampleWe have the set {1, 2, 3, 4, 5}. A subset of this is{1, 2, 3}. Another subset is {3, 4} or even another, {1}.However, {1, 6} is not a subset, since it contains anelement 6 which is not in the parent set.

In generalA is a subset of B if and only if every element of A is in B .

For ExampleWe have the set {1, 2, 3, 4, 5}.

A subset of this is{1, 2, 3}. Another subset is {3, 4} or even another, {1}.However, {1, 6} is not a subset, since it contains anelement 6 which is not in the parent set.

For ExampleWe have the set {1, 2, 3, 4, 5}. A subset of this is{1, 2, 3}.

Another subset is {3, 4} or even another, {1}.However, {1, 6} is not a subset, since it contains anelement 6 which is not in the parent set.

For ExampleWe have the set {1, 2, 3, 4, 5}. A subset of this is{1, 2, 3}. Another subset is {3, 4}

or even another, {1}.However, {1, 6} is not a subset, since it contains anelement 6 which is not in the parent set.

For ExampleWe have the set {1, 2, 3, 4, 5}. A subset of this is{1, 2, 3}. Another subset is {3, 4} or even another, {1}.

However, {1, 6} is not a subset, since it contains anelement 6 which is not in the parent set.

For ExampleWe have the set {1, 2, 3, 4, 5}. A subset of this is{1, 2, 3}. Another subset is {3, 4} or even another, {1}.However, {1, 6} is not a subset,

since it contains anelement 6 which is not in the parent set.

For ExampleLet A be all multiples of 4 and B be all multiples of 2. IsA a subset of B? And is B a subset of A?

Well, we can’t check every element in these sets, becausethey have an infinite number of elements. So we need toget an idea of what the elements look like in each, andthen compare them.

The sets areA = {. . . ,−8,−4, 0, 4, 8, . . .}B = {. . . ,−8,−6,−4,−2, 0, 2, 4, 6, 8, . . .}

For ExampleLet A be all multiples of 4 and B be all multiples of 2. IsA a subset of B? And is B a subset of A?Well, we can’t check every element in these sets, becausethey have an infinite number of elements. So we need toget an idea of what the elements look like in each, andthen compare them.

The sets areA = {. . . ,−8,−4, 0, 4, 8, . . .}B = {. . . ,−8,−6,−4,−2, 0, 2, 4, 6, 8, . . .}

The sets areA = {. . . ,−8,−4, 0, 4, 8, . . .}

B = {. . . ,−8,−6,−4,−2, 0, 2, 4, 6, 8, . . .}

The sets areA = {. . . ,−8,−4, 0, 4, 8, . . .}B = {. . . ,−8,−6,−4,−2, 0, 2, 4, 6, 8, . . .}

By pairing off members of the two sets, we can see thatevery member of A is also a member of B , but everymember of B is not a member of A.

A is a subset of B , but B is not a subset of A

By pairing off members of the two sets, we can see thatevery member of A is also a member of B ,

but everymember of B is not a member of A.

A is a subset of B ,

but B is not a subset of A

Proper SubsetsLet A be a set. Is every element in A an element in A?(Yes, I wrote that correctly.)

So doesn’t that mean thatA is a subset of A? This doesn’t seem very proper, doesit? We want our subsets to be proper. So we introduceproper subsets.

DefinitionA is a proper subset of B if and only if every element inA is also in B , and there exists at least one element in Bthat is not in A.

Proper SubsetsLet A be a set. Is every element in A an element in A?(Yes, I wrote that correctly.) So doesn’t that mean thatA is a subset of A?

This doesn’t seem very proper, doesit? We want our subsets to be proper. So we introduceproper subsets.

Proper SubsetsLet A be a set. Is every element in A an element in A?(Yes, I wrote that correctly.) So doesn’t that mean thatA is a subset of A? This doesn’t seem very proper, doesit?

We want our subsets to be proper. So we introduceproper subsets.

Proper SubsetsLet A be a set. Is every element in A an element in A?(Yes, I wrote that correctly.) So doesn’t that mean thatA is a subset of A? This doesn’t seem very proper, doesit? We want our subsets to be proper.

So we introduceproper subsets.

Proper SubsetsLet A be a set. Is every element in A an element in A?(Yes, I wrote that correctly.) So doesn’t that mean thatA is a subset of A? This doesn’t seem very proper, doesit? We want our subsets to be proper. So we introduceproper subsets.

DefinitionA is a proper subset of B if and only if

every element inA is also in B , and there exists at least one element in Bthat is not in A.

DefinitionA is a proper subset of B if and only if every element inA is also in B ,

and there exists at least one element in Bthat is not in A.

For Example{1, 2, 3} is a subset of {1, 2, 3},

but is not a propersubset of {1, 2, 3}.{1, 2, 3} is a proper subset of {1, 2, 3, 4} because theelement 4 is not in the first set.

Notice that if A is a proper subset of B , then it is also asubset of B .

Even More NotationWhen we say that A is a subset of B , we write A ⊆ B .Or we can say that A is not a subset of B by A * B(“A is not a subset of B”)

For Example{1, 2, 3} is a subset of {1, 2, 3}, but is not a propersubset of {1, 2, 3}.

{1, 2, 3} is a proper subset of {1, 2, 3, 4} because theelement 4 is not in the first set.

For Example{1, 2, 3} is a subset of {1, 2, 3}, but is not a propersubset of {1, 2, 3}.{1, 2, 3} is a proper subset of {1, 2, 3, 4}

because theelement 4 is not in the first set.

For Example{1, 2, 3} is a subset of {1, 2, 3}, but is not a propersubset of {1, 2, 3}.{1, 2, 3} is a proper subset of {1, 2, 3, 4} because theelement 4 is not in the first set.

Notice that if A is a proper subset of B ,

then it is also asubset of B .

Even More NotationWhen we say that A is a subset of B , we write A ⊆ B .

Or we can say that A is not a subset of B by A * B(“A is not a subset of B”)

Empty or Null SetAs an example, think of the set of pianokeys on a guitar.

“But wait!” you say,“There are no piano keys on a guitar!”And right you are. It is a set with noelements.

DefinitionA set which contains no element is known as the EmptySet (or Null Set).

NotationIt is represented by ∅ Or by { } (a set with no elements)

Empty or Null SetAs an example, think of the set of pianokeys on a guitar. “But wait!” you say,“There are no piano keys on a guitar!”

And right you are. It is a set with noelements.

Empty or Null SetAs an example, think of the set of pianokeys on a guitar. “But wait!” you say,“There are no piano keys on a guitar!”And right you are.

It is a set with noelements.

Empty or Null SetAs an example, think of the set of pianokeys on a guitar. “But wait!” you say,“There are no piano keys on a guitar!”And right you are. It is a set with noelements.

Set - DefinitionA set is a collection of well defined objects.

For ExampleYou could have a set made up of your ten best friends

Friends = { Anbu, Babu, John, Joel, Dass, David, Ravi,Raj, Selva, Vimal }

Anbu, Babu, Ravi and Raj play Soccer.

Selva, Ravi and Raj play Tennis.

UnionYou can now list your friends that play Soccer ORTennis.

This is called a “Union” of sets and has thespecial symbol

Soccer ∪ Tennis = { Anbu, Babu, Ravi, Raj, Selva }

Not everyone is in that set. Only your friends that playSoccer or Tennis (or both).

UnionYou can now list your friends that play Soccer ORTennis. This is called a “Union” of sets and has thespecial symbol

Not everyone is in that set.

Only your friends that playSoccer or Tennis (or both).

Intersection“Intersection” is when you have to be in BOTH sets.

Inour case that means they play both Soccer AND Tennis.Which is Ravi and Raj. The special symbol forIntersection is an upside down

⋃like this

⋂. And this is

how we write it down

Soccer ∩ Tennis = { Ravi, Raj }

Intersection“Intersection” is when you have to be in BOTH sets. Inour case that means they play both Soccer AND Tennis.

Which is Ravi and Raj. The special symbol forIntersection is an upside down

⋃like this

⋂. And this is

Intersection“Intersection” is when you have to be in BOTH sets. Inour case that means they play both Soccer AND Tennis.Which is Ravi and Raj.

The special symbol forIntersection is an upside down

⋃like this

⋂. And this is

Intersection“Intersection” is when you have to be in BOTH sets. Inour case that means they play both Soccer AND Tennis.Which is Ravi and Raj. The special symbol forIntersection is an upside down

⋃like this

And this ishow we write it down

⋃like this

⋂. And this is

⋃like this

⋂. And this is

DifferenceYou can also subtract one set from another.

Forexample, taking Soccer and subtracting Tennis meanspeople that play Soccer but NOT Tennis. Which is Anbuand Babu. And this is how we write it down

Soccer - Tennis = { Anbu, Babu }

DifferenceYou can also subtract one set from another. Forexample, taking Soccer and subtracting Tennis meanspeople that play Soccer but NOT Tennis.

Which is Anbuand Babu. And this is how we write it down

DifferenceYou can also subtract one set from another. Forexample, taking Soccer and subtracting Tennis meanspeople that play Soccer but NOT Tennis. Which is Anbuand Babu.

And this is how we write it down

DifferenceYou can also subtract one set from another. Forexample, taking Soccer and subtracting Tennis meanspeople that play Soccer but NOT Tennis. Which is Anbuand Babu. And this is how we write it down

Summary So Far

is Union: is in either set

is Intersection: must be in both sets

P − is Difference: in one set but not the other

Summary So Far

Origin of Set Theory

The basic ideas of set theory were developed by theGerman mathematician Georg Cantor (1845-1918).

He worked on certain kinds of infinite seriesparticularly on Fourier series

Most mathematicians accept set theory as a basis ofmodern mathematical analysis

Cantor’s work was fundamental to the laterinvestigation of Mathematical logic.

Definitions

SetA set is a collection of well-defined objects. The objectsof a set are called elements or members of the set.

The main property of a set in mathematics is that it iswell-defined. This means that given any object, it mustbe clear whether that object is a member (element) ofthe set or not. The objects of a set are all distinct, i.e.,no two objects are the same.

Definitions

The main property of a set in mathematics is that it iswell-defined.

This means that given any object, it mustbe clear whether that object is a member (element) ofthe set or not. The objects of a set are all distinct, i.e.,no two objects are the same.

Definitions

The main property of a set in mathematics is that it iswell-defined. This means that given any object, it mustbe clear whether that object is a member (element) ofthe set or not.

The objects of a set are all distinct, i.e.,no two objects are the same.

Definitions

The main property of a set in mathematics is that it iswell-defined. This means that given any object, it mustbe clear whether that object is a member (element) ofthe set or not. The objects of a set are all distinct, i.e.,no two objects are the same.

Definitions

ExampleWhich of the following collections are well - defined ?

(1) The collection of male students in our class.

(2) The collection of numbers 2, 4, 6, 10 and 12.

(3) The collection of states in India.

(4) The collection of all good movies.

(1), (2) and (3) are well-defined and therefore they aresets. (4) is not well-defined because the word good isnot defined. Therefore, (4) is not a set.

Definitions

(1), (2) and (3) are well-defined and therefore they aresets.

(4) is not well-defined because the word good isnot defined. Therefore, (4) is not a set.

Definitions

(1), (2) and (3) are well-defined and therefore they aresets. (4) is not well-defined because the word good isnot defined.

Therefore, (4) is not a set.

Definitions

Cardinal Numberthe number of elements in a set is called the cardinalnumber of the set.

The cardinal number of the set A isdenoted by n(A).

ExampleConsider the set A = {−1, 0, 1, 2, 3, 4, 5}. The set A has7 elements. ∴ The cardinal number of A is 7. i.e.,n(A) = 7.

Definitions

Cardinal Numberthe number of elements in a set is called the cardinalnumber of the set. The cardinal number of the set A isdenoted by n(A).

Definitions

ExampleConsider the set A = {−1, 0, 1, 2, 3, 4, 5}.

The set A has7 elements. ∴ The cardinal number of A is 7. i.e.,n(A) = 7.

Definitions

ExampleConsider the set A = {−1, 0, 1, 2, 3, 4, 5}. The set A has7 elements.

∴ The cardinal number of A is 7. i.e.,n(A) = 7.

Definitions

Empty SetA set containing no elements is called the empty set ornull set or void set.

ExampleConsider the set A = {x : x < 1, x ∈ N}. There are nonatural numbers which are less than 1.

Definitions

ExampleConsider the set A = {x : x < 1, x ∈ N}.

There are nonatural numbers which are less than 1.

Definitions

ExampleConsider the set A = {x : x < 1, x ∈ N}. There are nonatural numbers which are less than 1.

Definitions

Finite SetIf the number of elements in a set is zero or finite, thenthe set is called a finite set.

Example

P Consider the set A of natural numbers between 8 and9. There is no natural numbers between 8 and 9.A = { } and n(A) = 0. ∴ A is finite set.

P Consider the set X = {x : x is an integer and−1 ≤ x ≤ 2}. X = {−1, 0, 1, 2} and n(X ) = 4. ∴ Xis a finite set.

Definitions

Example

P Consider the set A of natural numbers between 8 and9.

There is no natural numbers between 8 and 9.A = { } and n(A) = 0. ∴ A is finite set.

Definitions

Example

P Consider the set A of natural numbers between 8 and9. There is no natural numbers between 8 and 9.

A = { } and n(A) = 0. ∴ A is finite set.

Definitions

Example

P Consider the set A of natural numbers between 8 and9. There is no natural numbers between 8 and 9.A = { } and n(A) = 0.

∴ A is finite set.

Definitions

Example

Definitions

Example

P Consider the set X = {x : x is an integer and−1 ≤ x ≤ 2}.

X = {−1, 0, 1, 2} and n(X ) = 4. ∴ Xis a finite set.

Definitions

Example

P Consider the set X = {x : x is an integer and−1 ≤ x ≤ 2}. X = {−1, 0, 1, 2} and n(X ) = 4.

∴ Xis a finite set.

Definitions

Example

Definitions

Infinite SetA set is said to be an infinite set if the number ofelements in the set is not finite.

ExampleLet W = The set of all whole numbers, i.e.,W = {0, 1, 2, 3, . . .}. The set of whole numbers containinfinite number of elements. ∴ W is an infinite set.

Definitions

ExampleLet W = The set of all whole numbers,

i.e.,W = {0, 1, 2, 3, . . .}. The set of whole numbers containinfinite number of elements. ∴ W is an infinite set.

Definitions

ExampleLet W = The set of all whole numbers, i.e.,W = {0, 1, 2, 3, . . .}.

The set of whole numbers containinfinite number of elements. ∴ W is an infinite set.

Definitions

ExampleLet W = The set of all whole numbers, i.e.,W = {0, 1, 2, 3, . . .}. The set of whole numbers containinfinite number of elements.

∴ W is an infinite set.

Definitions

ExampleLet W = The set of all whole numbers, i.e.,W = {0, 1, 2, 3, . . .}. The set of whole numbers containinfinite number of elements. ∴ W is an infinite set.

Definitions

Singleton Seta set containing only one element is called a singleton set

ExampleConsider the set A = {x : x is an integer and1 < x < 3}. A = {2}. i.e., A has only one element. ∴ Ais a singleton set.

Definitions

ExampleConsider the set A = {x : x is an integer and1 < x < 3}.

A = {2}. i.e., A has only one element. ∴ Ais a singleton set.

Definitions

ExampleConsider the set A = {x : x is an integer and1 < x < 3}. A = {2}. i.e., A has only one element.

∴ Ais a singleton set.

Definitions

ExampleConsider the set A = {x : x is an integer and1 < x < 3}. A = {2}. i.e., A has only one element. ∴ Ais a singleton set.

Definitions

Equivalent Settwo sets A and B are said to be equivalent if they havethe same number of elements.

In other words, A and Bare equivalent if n(A) = n(B).

ExampleConsider the sets A = {7, 8, 9, 10} and B = {3, 5, 6, 11}.Here, n(A) = 4 and n(B) = 4. ∴ A ≡ B .

Definitions

Equivalent Settwo sets A and B are said to be equivalent if they havethe same number of elements. In other words, A and Bare equivalent if n(A) = n(B).

Definitions

ExampleConsider the sets A = {7, 8, 9, 10}

and B = {3, 5, 6, 11}.Here, n(A) = 4 and n(B) = 4. ∴ A ≡ B .

Definitions

ExampleConsider the sets A = {7, 8, 9, 10} and B = {3, 5, 6, 11}.

Here, n(A) = 4 and n(B) = 4. ∴ A ≡ B .

Definitions

ExampleConsider the sets A = {7, 8, 9, 10} and B = {3, 5, 6, 11}.Here, n(A) = 4 and n(B) = 4.

∴ A ≡ B .

Definitions

Equal SetTwo sets A and B are said to be equal if they containexactly the same elements, regardless of order.

Otherwise the sets are said to be unequal. In otherwords, two sets A and B , are said to be equal if

1 every element of A is also an element of B and2 every element of B is also an element of A.

ExampleConsider the sets A = {a, b, c , d} and B = {d , b, a, c}.Set A and B contain exactly the same elements.∴ A = B .

Definitions

Equal SetTwo sets A and B are said to be equal if they containexactly the same elements, regardless of order.Otherwise the sets are said to be unequal.

In otherwords, two sets A and B , are said to be equal if

Definitions

Equal SetTwo sets A and B are said to be equal if they containexactly the same elements, regardless of order.Otherwise the sets are said to be unequal. In otherwords, two sets A and B , are said to be equal if

Definitions

1 every element of A is also an element of B and

2 every element of B is also an element of A.

Definitions

ExampleConsider the sets A = {a, b, c , d} and B = {d , b, a, c}.

Set A and B contain exactly the same elements.∴ A = B .

Definitions

ExampleConsider the sets A = {a, b, c , d} and B = {d , b, a, c}.Set A and B contain exactly the same elements.

∴ A = B .

Definitions

SubsetA set A is a subset of set B if every element of A is alsoan element of B .

In symbol we write A ⊆ B

� Read A ⊆ B as ‘A is a subset of B ’ or ‘A iscontained in B ’

� Read A * B as ‘A is not a subset of B ’ or ‘A is notcontained in B ’

Definitions

SubsetA set A is a subset of set B if every element of A is alsoan element of B . In symbol we write A ⊆ B

Definitions

ExampleConsider the sets A = {7, 8, 9} and B = {7, 8, 9, 10}.

We see that every element of A is also an element of B .∴ A is a subset of B . i.e., A ⊆ B .

Definitions

ExampleConsider the sets A = {7, 8, 9} and B = {7, 8, 9, 10}.We see that every element of A is also an element of B .

∴ A is a subset of B . i.e., A ⊆ B .

Definitions

ExampleConsider the sets A = {7, 8, 9} and B = {7, 8, 9, 10}.We see that every element of A is also an element of B .∴ A is a subset of B . i.e., A ⊆ B .

Definitions

Proper SubsetA set A is said to be a proper subset of set B if A ⊆ Band A 6= B .

In symbol we write A ⊂ B . B is called superset of A.

ExampleConsider the sets A = {5, 7, 8} and B = {5, 6, 7, 8}.Every element of A is also an element of B and A 6= B .∴ A is a proper subset of B .

Definitions

Proper SubsetA set A is said to be a proper subset of set B if A ⊆ Band A 6= B . In symbol we write A ⊂ B .

B is called superset of A.

Definitions

Proper SubsetA set A is said to be a proper subset of set B if A ⊆ Band A 6= B . In symbol we write A ⊂ B . B is called superset of A.

Definitions

ExampleConsider the sets A = {5, 7, 8} and B = {5, 6, 7, 8}.

Every element of A is also an element of B and A 6= B .∴ A is a proper subset of B .

Definitions

ExampleConsider the sets A = {5, 7, 8} and B = {5, 6, 7, 8}.Every element of A is also an element of B and A 6= B .

∴ A is a proper subset of B .

Definitions

Power SetThe set of all subsets of A is said to be the power set ofthe set A.

The power set of a set A is denoted by ρ(A).

K The number of subsets of a set with m elements is 2m

K The number of proper subsets of a set with melements is 2m − 1

ExampleLet A = {−3, 4}. The subsets of A areφ, {−3}, {4}, {−3, 4}. Then the power set of A isρ(A) =

{φ, {−3}, {4}, {−3, 4}

Definitions

Power SetThe set of all subsets of A is said to be the power set ofthe set A. The power set of a set A is denoted by ρ(A).

{φ, {−3}, {4}, {−3, 4}

Definitions

{φ, {−3}, {4}, {−3, 4}

Definitions

{φ, {−3}, {4}, {−3, 4}

Definitions

ExampleLet A = {−3, 4}.

The subsets of A areφ, {−3}, {4}, {−3, 4}. Then the power set of A isρ(A) =

{φ, {−3}, {4}, {−3, 4}

Definitions

ExampleLet A = {−3, 4}. The subsets of A areφ, {−3}, {4}, {−3, 4}.

Then the power set of A isρ(A) =

{φ, {−3}, {4}, {−3, 4}

Definitions

{φ, {−3}, {4}, {−3, 4}

}J.Maria Joseph Ph.D., SJC, Trichy-2. Set Theory July 1, 2015 43 / 62

Definitions

Universal SetThe set that contains all the elements underconsideration in a given discussion is called the universalset.

The universal set is denoted by U .

ExampleIf the elements currently under discussion are integers,then the universal set U is the set of all integers. i.e.,U = {x : x ∈ Z}.

Definitions

Universal SetThe set that contains all the elements underconsideration in a given discussion is called the universalset. The universal set is denoted by U .

Definitions

ExampleIf the elements currently under discussion are integers,

then the universal set U is the set of all integers. i.e.,U = {x : x ∈ Z}.

Definitions

ExampleIf the elements currently under discussion are integers,then the universal set U is the set of all integers.

i.e.,U = {x : x ∈ Z}.

Definitions

Complement SetThe set of all elements of U (universal set) that are notelements of A ⊆ U is called the complement of A.

Thecomplement of A is denoted by A′ or Ac .

ExampleLet U = {a, b, c , d , e, f , g , h} and A = {b, d , g , h}.Then A′ = {a, c , e, f }

Definitions

Complement SetThe set of all elements of U (universal set) that are notelements of A ⊆ U is called the complement of A. Thecomplement of A is denoted by A′ or Ac .

Definitions

ExampleLet U = {a, b, c , d , e, f , g , h} and A = {b, d , g , h}.

Then A′ = {a, c , e, f }

Definitions

Union of SetsThe union of two sets A and B is the set of elementswhich are in A or in B or in both A and B .

We write theunion of sets A and B as A ∪ B .

In symbol, A ∪ B = {x : x ∈ A or x ∈ B}

ExampleLet A = {11, 12, 13, 14} and B = {9, 10, 12, 14, 15}.Then A ∪ B = {9, 10, 11, 12, 13, 14, 15}.

Definitions

Union of SetsThe union of two sets A and B is the set of elementswhich are in A or in B or in both A and B . We write theunion of sets A and B as A ∪ B .

Definitions

ExampleLet A = {11, 12, 13, 14} and B = {9, 10, 12, 14, 15}.

Then A ∪ B = {9, 10, 11, 12, 13, 14, 15}.

Definitions

Intersection of SetsThe intersection of two sets A and B is the set of allelements common to both A and B .

We write theinteresction of sets A and B as A ∩ B .

In symbol, A ∩ B = {x : x ∈ A and x ∈ B}

ExampleLet A = {11, 12, 13, 14} and B = {9, 10, 12, 14, 15}.Then A ∩ B = {12, 14}.

Definitions

Intersection of SetsThe intersection of two sets A and B is the set of allelements common to both A and B . We write theinteresction of sets A and B as A ∩ B .

Definitions

ExampleLet A = {11, 12, 13, 14} and B = {9, 10, 12, 14, 15}.

Then A ∩ B = {12, 14}.

Definitions

Disjoint SetsTwo sets A and B are said to be disjoint if there is noelement common to both A and B .

In other words, if Aand B are disjoint sets, then A ∩ B = ∅.

ExampleConsider the sets A = {5, 6, 7, 8} and B = {11, 12, 13}.We have A ∩ B = φ. So A and B are disjoint sets.

Definitions

Disjoint SetsTwo sets A and B are said to be disjoint if there is noelement common to both A and B . In other words, if Aand B are disjoint sets, then

A ∩ B = ∅.

Definitions

Disjoint SetsTwo sets A and B are said to be disjoint if there is noelement common to both A and B . In other words, if Aand B are disjoint sets, then A ∩ B = ∅.

Definitions

ExampleConsider the sets A = {5, 6, 7, 8} and B = {11, 12, 13}.

We have A ∩ B = φ. So A and B are disjoint sets.

Definitions

ExampleConsider the sets A = {5, 6, 7, 8} and B = {11, 12, 13}.We have A ∩ B = φ.

So A and B are disjoint sets.

Definitions

Difference of two SetsThe difference of the two sets A and B is the set of allelements belonging to A but not to B .

The difference ofthe two sets is denoted by A− B

= In symbol, A− B = {x : x ∈ A and x /∈ B}= Similarly, B − A = {x : x ∈ B and x /∈ A}

ExampleConsider the set A = {2, 3, 5, 7, 11} andB = {5, 7, 9, 11, 13}. To find A− B , we remove theelements of B from A. ∴ A− B = {2, 3}.

Definitions

Difference of two SetsThe difference of the two sets A and B is the set of allelements belonging to A but not to B . The difference ofthe two sets is denoted by A− B

Definitions

= In symbol, A− B = {x : x ∈ A and x /∈ B}

= Similarly, B − A = {x : x ∈ B and x /∈ A}

Definitions

ExampleConsider the set A = {2, 3, 5, 7, 11} andB = {5, 7, 9, 11, 13}.

To find A− B , we remove theelements of B from A. ∴ A− B = {2, 3}.

Definitions

ExampleConsider the set A = {2, 3, 5, 7, 11} andB = {5, 7, 9, 11, 13}. To find A− B ,

we remove theelements of B from A. ∴ A− B = {2, 3}.

Definitions

ExampleConsider the set A = {2, 3, 5, 7, 11} andB = {5, 7, 9, 11, 13}. To find A− B , we remove theelements of B from A.

∴ A− B = {2, 3}.

Definitions

Symmetric DifferenceThe symmetric difference of two sets A and B is theunion of their differences

and is denoted by A∆B .

Thus, A∆B = (A− B) ∪ (B − A)

ExampleConsider the sets A = {a, b, c , d} and B = {b, d , e, f }.We have, A− B = {a, c} and B − A = {e, f }.∴ A∆B = {(A− B) ∪ (B − A) = {a, c , e, f }.

Definitions

Symmetric DifferenceThe symmetric difference of two sets A and B is theunion of their differences and is denoted by A∆B .

Thus, A∆B = (A− B) ∪ (B − A)

Definitions

Thus, A∆B = (A− B) ∪ (B − A)

Definitions

Thus, A∆B = (A− B) ∪ (B − A)

ExampleConsider the sets A = {a, b, c , d} and B = {b, d , e, f }.

We have, A− B = {a, c} and B − A = {e, f }.∴ A∆B = {(A− B) ∪ (B − A) = {a, c , e, f }.

Definitions

Thus, A∆B = (A− B) ∪ (B − A)

ExampleConsider the sets A = {a, b, c , d} and B = {b, d , e, f }.We have, A− B = {a, c}

and B − A = {e, f }.∴ A∆B = {(A− B) ∪ (B − A) = {a, c , e, f }.

Definitions

Thus, A∆B = (A− B) ∪ (B − A)

ExampleConsider the sets A = {a, b, c , d} and B = {b, d , e, f }.We have, A− B = {a, c} and B − A = {e, f }.

∴ A∆B = {(A− B) ∪ (B − A) = {a, c , e, f }.

Definitions

Thus, A∆B = (A− B) ∪ (B − A)

Definitions

For any two finite sets A and B , we have

P n(A) = n(A− B) + n(A ∩ B)

P n(B) = n(B − A) + n(A ∩ B)